Jun 17, 2017 - and we denote it by dimp(G). We begin with a definition. Definition 2.1. Let n ⥠2. The n-regular rank of G is the order-type of the following ...
arXiv:1706.05471v1 [math.LO] 17 Jun 2017
Strong ordered Abelian groups and dp-rank Rafel Farr´e Universitat Polit`ecnica de Catalunya Departament de Matem`atiques C/Jordi Girona, 1-3 Edifici Omega, E-08034 Barcelona, Spain June 20, 2017 Abstract We provide an algebraic characterization of strong ordered Abelian groups: An ordered Abelian group is strong iff it has bounded regular rank and almost finite dimension. Moreover, we show that any strong ordered Abelian group has finite Dp-rank. We also provide a formula that computes the exact valued of the Dprank of any ordered Abelian group. In particular characterizing those ordered Abelian groups with Dp-rank equal to n. We also show the Dprank coincides with the Vapnik-Chervonenkis density.
1
Contents 1 Introduction 1.1 notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 4
2 A Quantifier Elimination for ordered Abelian groups with bounded regular rank 6 3 On computing dp-rank
11
4 dp-minimal ordered groups
17
5 Chain conditions
20
6 Strong ordered Abelian groups
24
7 Computing the dp-rank in ordered Abelian groups
27
8 VC-density of ordered Abelian groups
32
9 Gurevich-Schmitt Quantifier Elimination for ordered Abelian groups 34
2
1
Introduction
S. Shelah asked the question of which ordered Abelian groups are strongly dependent (a strong form of NIP). H. Adler introduced the notion of strong theory (a strong form of NTP2 ) which is a generalization of strongly dependent and coincides with the former in NIP theories. In other words, a theory (or more generally a type) is strongly dependent iff it is stong and NIP. Since it is wellknown that all ordered Abelian groups are NIP, the question can be stated as: which ordered Abelian groups are strong? One of the main results of the paper is Theorem 6.5, a characterization of strong ordered Abelian groups showing that an ordered Abelian group is strong iff it has bounded regular rank and almost finite dimension. See sections 2 for the definition of bounded regular rank and section 6 for the definition of almost finite dimension. As a corollary we obtain that any strong ordered Abelian group has finite dp-rank. Moreover we provide a formula that computes the exact value of the dp-rank. The paper is organized as follows. In section 2 we state and prove Theorem 2.4, a Quantifier Elimination for ordered Abelian groups with bounded regular rank. This QE will be applied in sections 4 and 6. We use a relative QE result of Gurevich and Schmitt to prove our result. The notation and statement of the relative QE are in section 9. In section 3 we introduce a variation of ict-patterns by relaxing a little bit the conditions on an ict-pattern. This will be useful to obtain upper bounds of dp-rank in Theorem 6.4. We call this new patterns wict-patterns (weak independence partition pattern). We will see that wict-patterns compute dp-rank. This is the content of Proposition 3.5. The nice point is that when there is QE in the language it is enough to consider wict-patterns constituted by literals (atomic or negation of atomic formulas). This is basically the content of Propositions 3.6 and 3.7. Ict-patterns do not have this property since Proposition 3.7 fails for ict-patterns. We also show in Proposition 3.13 that directed families of formulas cannot occur in two different rows of the same pattern. this is also very useful in order to provide upper bounds on the dp-rank. In section 4 we provide a characterization of dp-minimal ordered Abelian groups. The arguments of this section are the same as those in sections 6. Moreover the results here are a corollary as those in section 6. This section is just a warming-up of section 6. The algebraic characterization of a dp-minimal ordered Abelian group as having finite dimension has been first published in [13] using another terminology. In section 5 we prove Propositions 5.4 and 5.5. They may be considered as chain conditions Theorems, see [14] for close results. They are useful in proving lower bounds for burden (and dp-rank). In particular 5.5 is useful to obtain consequences from strongness. It is used in 6.1 and 6.2. Propositions 5.4 is used in section 7. 3
Section 6 contains Theorem 6.5, the characterization of strong ordered Abelian groups. It also contains the proof that all strong ordered Abelian groups have finite dp-rank. Section 7 is a refinement of the arguments of section 6 to obtain a formula in Theorem 7.2 that computes the exact value of the dp-rank. The characterisation of Strong ordered Abelian groups as having bounded regular rank and almost finite dimension is obtained independently in [12] using another terminology and different proof techniques. The authors also prove in [12] that any strong ordered Abelian group has finite dp-rank. They also provide a formula similar to 7.2. Corollary 7.6 is also there. Section 8 contains a proof that for ordered Abelian groups, the VC-density coincides with the dp-rank. More precisely Theorem 8.3 shows that the dp-rank coincides with the VC-density function evaluated at 1. Most of the results of this paper (sections 2 , 3, 4 ,5, 6 and 7 )were obtained in 2014, but they remained unpublished until now. They were exposed in detail in the Barcelona Model theory seminar along various sessions during March and April 2014 (see http://www.ub.edu/modeltheory/mt.htm). I also gave a talk about the subject in the Mathematical Logic Seminar in Freiburg the 25th. of June 2014 (see http://logik.mathematik.uni-freiburg.de/lehre/archiv/ ss14/oberseminar-ss14.html).
1.1
notation
We use x to denote a tuple of variables. Most of the time is finite but sometimes could be infinite. We denote by |x| the length of the tuple. We use x to denote a single variable. The same conventional notation will be used for tuples of elements. We will denote by P the set of all prime numbers. We always play with the monster model of some complete theory T . But in fact a little bit of saturation (a weakly saturated model) is enough. The notation an terminology specific for ordered Abelian groups is contained in sections 2 and 9. When we say that a structure is VCA-minimal, dp-minimal, strong (or in general any abstract property of theories) we mean the theory of the structure is VCA-minimal, dp-minimal, strong... In practice we always may assume the structure is as much saturated as needed (by replacing it by another elementarily equivalent). The same applies to invariants of a theory, like dp-rk(T ), bdn(T )... In this case we denote dp-rk(M ) = dp-rk(Th(M )) and similarly by bdn and VC-density. When dealing with dp-rank and burden there two possible ways to define them, both common in the literature. The first possibility is to define the dprank of a type as the supremum of all depths of all ict-patterns for the type. The second one is Shelah’s style: the the dp-rank of a type is the first cardinal κ for which there is no ict-patterns for the type of depth κ (and ∞ if such cardinal does not exists). We choose the first definition, although we do not use subscripts to distinguish whether those supremums are achieved or not. 4
If ϕ(x, y) is a formula, M is a model and b is a tuple in M of the same length as y, we will use ϕ(M, b) to denote the set defined by ϕ(x, b) in M , namely a ∈ M |x| | M |= ϕ(a, b) . If G is an ordered Abelian group we will denote ∆ G to indicate that ∆ is a convex subgroup of G and ∆ G that ∆ is a proper convex subgroup.
5
2
A Quantifier Elimination for ordered Abelian groups with bounded regular rank
In this section we generalize a quantifier elimination result of W. Weispfenning in [24]. Weispfenning result is precisely our theorem 2.4 in the case when G has finite regular rank, see the comments after proposition 2.3. We deduce our result from a more general QE of Y. Gurevich and P. Schmitt, see [10], [18], [17]. This more general result, the notation and the terminology is explained in section 9 of this paper. If p is a prime number, G/pG is a vector space over the field with p elements. We call the dimension of this vector space the p-dimension of G and we denote it by dimp (G). We begin with a definition. Definition 2.1. Let n ≥ 2. The n-regular rank of G is the order-type of the following ordered set: (RJn (G), ⊆) , where RJn (G) denotes {An (g) | g ∈ G − {0}}, the set of all n-regular jumps. The n-regular rank of an ordered Abelian group G is a linear order. When it is finite we can identify this order with its cardinal. Moreover it can be easily characterized without any reference to the sets An (g) as next remark shows. Remark 2.2.
1. G has n-regular rank 0 iff G = {0}.
2. G has n-regular rank 1 iff G is n-regular and non-trivial. 3. G has n-regular rank equal to m iff there are ∆0 , . . . , ∆m convex subgroups of G, such that: (a) {0} = ∆0 ∆1 · · · ∆m = G (b) ∆i+1 /∆i is n-regular for 0 ≤ i < m (c) ∆i+1 /∆i is not n-divisible for 0 < i < m. In this case RJn (G) = {∆0 , . . . , ∆m−1 }. 4. It G has finite n-regular rank and H ≡ G then H has the same n-regular rank as G. Hence, when finite, the n-regular rank is an invariant of the theory of G. If G has infinite n-regular rank, the regular rank of a κsaturated model of the theory of G is a linear order which has cardinality at least κ. It could be interesting characterize the n-regular rank of the monster model (it is not an ηα -ordered set in general). 5. If dimp (G) is finite then the p-regular rank of G is at most dimp (G) + 1. 6. The number of convex subgroups ∆ of G with G/∆ discrete is bounded by the n-regular rank for each n. In particular, if G has finite n-regular rank for some n, then the number of convex subgroups ∆ of G with G/∆ discrete is finite. 6
7. If the n-regular rank of G is finite then each convex subgroup of the form Fn (x) belongs to RJn (G). Proof. 1 and 2 are particular cases of 3, and 5 follows from 3. 3. If we have such a chain then RJn (G) = {∆0 , . . . , ∆m−1 }, since g ∈ ∆i+1 − ∆i implies An (g) = ∆i . 4. G has n-regular rank m iff in G holds: m−1 m−2 _ ^ ∃y0 , . . . , ym−1 ∀z An (z) = An (yj ) ∧ An (yj ) ⊂ An (yj+1 ) j=0
j=0
6. Because G/∆ discrete with first positive element 1∆ implies ∆ = An (1∆ ) and thus ∆ ∈ RJn (G) for any n ≥ 2. 7. Assume RJn (G) = {∆0 , . . . , ∆m−1 } as in 3 and ∆i Fn (x) ∆i+1 . Then x ∈ ∆i+1 + nG. By regularity of the jump, ∆i+1 /Fn (x) is n-divisible, thus ∆i+1 = Fn (x) + n∆i+1 . Then x ∈ Fn (x) + nG, a contradiction. S We also denote by RJ(G) the set n≥2 RJn (G) an call it the set of regular jumps of G. Proposition 2.3. Let G be an ordered Abelian group. The following are equivalent: 1. G has finite p-regular rank for each prime p. 2. G has finite n-regular rank for each n ≥ 2. 3. There is some cardinal κ such that for any H ≡ G, |RJ(H)| ≤ κ (RJ(H) is countable). 4. For any H ≡ G, any definable convex subgroup of H has a definition without parameters. 5. There is some cardinal κ such that for any H ≡ G, H has at most κ (countably many) definable convex subgroups. Moreover, in this case, RJ(G) is the collection of all proper definable convex subgroups of G and all are definable without parameters. S S Proof. 1 ⇒2. RJn (G) ⊆ p|n RJp (G) because An (g) = p|n Ap (g). 2 ⇒3. Let H ≡ G. By Point 4 in Remark 2.2 H has also finite n-regular rank for each n so RJ(H) is countable. 3 ⇒4 Let H ≡ G. A compactness argument shows that H must have finite n-regular rank for each n. By Theorem 4.1 of [6] any proper definable convex subgroup of H is an intersection of elements of RJn (H) for some n. The finiteness of RJn (H) implies that RJ(H) is the set of all proper definable convex
7
subgroups of H. Now, if RJn (H) = {∆0 , . . . , ∆m−1 } and ∆i = An (gi ) then ∆i can be defined in H by the formula m−2 ^ ∃y0 , . . . , ym−1 x ∈ An (yi ) ∧ An (yj ) ⊂ An (yj+1 ) j=0
4 ⇒5 is obvious. 5 ⇒1 If for some p, RJp (G) is infinite, a compactness argument allows us to find H ≡ G where RJp (H) is as bigSas we want. Finally observe that RJn (G) = p∈P RJp (G). In view of point 3 of Proposition 2.3 will say that an ordered Abelian group has bounded regular rank if it satisfies any of the conditions of Proposition 2.3. By Facts 9.2.7, RJn = Spn when RJn is finite. Therefore an ordered Abelian group has bounded regular rank iff all the spines Spn (G) are finite. In this case we define the regular rank of G as |RJ(G)|, the cardinality of RJ(G). It is either finite or ℵ0 . Observe that the property of having finite or bounded regular rank depends only in the theory of G. Also the value of the regular rank depends only on the theory (we may say that the regular rank is ∞ or non-defined when the group has not bounded regular rank). One can easily check that Weispfenning QE in [24] is just Theorem 2.4 in the case of finite regular rank. Theorem 2.4. Let G be an ordered Abelian group with bounded regular rank. Then G admits QE in the following language: L = {+, −, 0, ≤} ∪ {1∆ | ∆ ∈ RJ(G), G/∆ discrete} ∪ {x ≡ y
mod ∆ | ∆ ∈ RJ(G)} ∪
{x ≡ y
mod (∆ + pm G) | p ∈ P, ∆ ∈ RJp (G), m ≥ 1}
Here, if G/∆ is discrete, 1∆ denotes an element of G whose projection to G/∆ is the smallest positive element. Proof. We begin by remarking that we can add the following predicates for free: {x ≡ y
mod (∆ + nG) | ∆ ∈ RJ(G), n ≥ 1}
In fact we could have added any set of predicates of the form x≡y
mod (∆ + nG)
even if ∆ is not definable. This is because: Claim 2.5.
1. If n = pr11 · · · prkk then x≡y
mod (∆ + nG) ⇐⇒
^ i
8
x≡y
mod (∆ + pri i G)
2. If ∆ ∈ / RJp (G) ∪ {G}, let ∆1 , ∆2 be consecutive elements of RJp (G) ∪ {G} such that ∆1 ⊂ ∆ ⊂ ∆2 . Then mod (∆ + pr G) ⇐⇒ x ≡ y
x≡y
mod (∆2 + pr G)
Proof of the claim: 1 follows from the following formula: t \
∆ + mi G = ∆ + lcm(m1 , . . . , mt )G.
i=1
To prove this it is enough to prove (∆+nG)∩(∆+mG) = ∆+lcm(n, m)G. The other inclusion being obvious, it suffices to prove ⊆. Assume a = δ1 + mg1 = δ2 + ng2 with δi ∈ ∆ and gi ∈ G. Denote d = gcd(m, n), m = dm0 , n = dn0 . Let 1 = λm0 +µn0 be a B´ezout identity for n0 , m0 . Then a = λm0 a+µn0 a = λm0 (δ2 + ng2 )+µn0 (δ1 +mg1 ) = (λm0 δ2 +µn0 δ1 )+lcm(n, m)(λg2 +µg1 ) ∈ ∆+lcm(n, m)G. 2 follows because ∆2 /∆ p-divisible implies ∆2 = ∆ + pr ∆2 and thus ∆ + r p G = ∆2 + pr G. This ends the proof of the claim. We will use theorem 9.4. Let n be as in the statement of 9.4. Since RJn (G) is the domain of Spn (G) and is finite , Spn (G) |= ψ0 (C1 , . . . , Cm , D1 , . . . , Dr ) is equivalent to a boolean combination of the S following formulas An (ti (g)) = ∆ and Fn (si (g)) = ∆ with ∆ ∈ RJn (G) ⊆ p|n RJp (G). The following claim shows that this has a quantifier-free translation into the language L: Claim 2.6.
1. An (x) = ∆ ⇐⇒ (x 6≡ 0
mod ∆) ∧ (x ≡ 0
mod ∆n+ )
2. Fn (x) = ∆ ⇐⇒ (x 6≡ 0
mod (∆ + nG)) ∧ (x ≡ 0
mod (∆n+ + nG)),
where ∆n+ denotes the successor of ∆ in RJn (G). It remains to prove that the LOG∗ -predicates Mk , E(n,k) and D(p,r,i) are expressible without quantifiers in L. This is is done in the next claim. This ends the proof of the theorem. _ Claim 2.7. 1. Mk (x) ⇐⇒ x ≡ k1∆ mod ∆ G/∆ discrete
2. E(n,k) (x) ⇐⇒
_
x ≡ 0 mod (∆n+ + nG) ∧ (x ≡ k1∆ mod (∆ + nG))
G/∆ discrete
3. D(p,r,i) (x) ⇐⇒ (x ≡ 0 mod pr G) ∨
_
x ≡ 0 mod (∆ + pi G) ∧
G/∆ discrete p+
x ≡ 0 mod (∆
+ p G) ∧ (x 6≡ 0 mod (∆ + pr G)) r
9
Proof of the claim: 1 is easy, lets see 2. By remark 7 in 2.2, assume Fn (x) = ∆ for some ∆ ∈ RJn (G). It is easy to see that in this case Γ1,n (x) = ∆ + nG and Γ2,n (x) = ∆n+ + nG. Moreover, if G/∆ is discrete, [x] = k [1∆ ] in Γn (x) iff x ≡ k1∆ mod ∆ + nG. Now, keeping in mind that Fn (x) = ∆ is equivalent to (x ≡ 0 mod ∆n+ + nG) ∧ (x 6≡ 0 mod ∆ + nG), we get the result. One must bear in mind also that, since k > 0, x ≡ k1∆ mod ∆ + nG implies (x 6≡ 0 mod ∆ + nG). To prove 3, assume Fpr (x) = ∆ for some ∆ ∈ RJpr (G). As before, Γ1,pr (x) = r r ∆ + pr G and Γ2,pr (x) = ∆p + pr G. Then [x] ∈ pi Γpr (x) iff x ∈ ∆ + pi ∆p + + r r pr G = (∆ + pi G) ∩ (∆p + + pr G). Moreover, ∆p + = ∆p+ , since for ordered Abelian groups, being p-regular (p-divisible) is equivalent to being pr -regular (pr -divisible). This ends the proof of the claim.
10
3
On computing dp-rank
Along this section we work again with the monster model of some complete theory T . We begin by recalling the definition of ict-pattern and dp-rank. The following was defined in [23] and [16]. Definition 3.1. Let p(x) be a partial type. An ict-pattern for p(x) consist in the following data: a sequence of formulas S := (ϕ(x, y i ) | i ∈ k) and an array A := (aij | i ∈ κ, j ∈ O) of parameters, where κ is a cardinal number and O is an infinite linearly ordered set such that: for every f ∈ Oκ , the following set of formulas is consistent with p(x): n o ΓSf (A) := ϕi (x, aif (i) ) | i ∈ κ ∪ ¬ϕi (x, aij ) | i ∈ κ, j 6= f (i) The cardinal number κ is called the depth of the pattern and we allow κ to be finite. We also say that it is an ict-pattern of type κ × O with sequence of formulas S and array A. The dp-rank of p(x), denoted by dp-rk(p(x)) is the supremum of all depths of all ict-patterns for p(x). If T denotes a complete theory with a main sort 1 the dp-rank of T , denoted by dp-rk(T ) is dp-rk(x = x) where x is a single variable for the main sort. Now we introduce wict-patterns. Definition 3.2. Let p(x) be a partial type. A wict-pattern for p(x) consist in the following data: a sequence of formulas S := (ϕ(x, y i ) | i ∈ k) and an array A := (aij | i ∈ κ, j ∈ O), where κ is a cardinal number and O is an infinite linearly ordered set such that: for every f ∈ Oκ , the following set of formulas is consistent with p(x): n o ∆Sf (A) := ϕi (x, aif (i) ) | i ∈ κ ∪ ¬ϕi (x, aij ) | i ∈ κ, j > f (i) The cardinal number κ is called the depth of the pattern and we allow κ to be finite. We also say that it is a wict-pattern of type κ × O with sequence of formulas S and array A. Remark 3.3. 1. By the same arguments as for ict-patterns, we can replace the parameters of a wict-pattern (without changing the sequence of formulas nor the depth) by another array with mutually indiscernible rows over the set of parameters of the type. This means that each row is indiscernible over the set of parameters of the type plus all other rows. 2. When the wict-pattern has mutually indiscernible rows over the set of parameters of p, it is enough to check the consistency of a single path: If p(x) ∪ ∆Sf (A)(x) is consistent for some f ∈ Oκ then the same holds for any f ∈ Oκ . 1 there is a sort, called main sort, such that all other sorts are obtained as sorts of imaginaries of the theory of the main sort alone
11
3. The previous fact is not true in general for ict-patterns, it depends on the order-type of O. For instance, it holds for ict-patterns of kind κ × Z, but not for ict-patters of kind κ × ω (as the formula x > y shows in the theory on dense linear order without endpoints). The following kind of patterns were also first used by Shelah. The name ‘special’ is not standard and is simply used to distinguish them from the other two. Definition 3.4. Let p(x) be a partial type. A special pattern for p(x) consist in the following data: a sequence of formulas S := (ϕ(x, y i ) | i ∈ k) and an array A := (aij | i ∈ κ, j ∈ ω) with mutually indiscernible rows, where κ is a cardinal number such that the following set of formulas is consistent: (3.1) p(x) ∪ ϕi (x, ai0 ) | i ∈ κ ∪ ¬ϕi (x, ai1 ) | i ∈ κ The following shows we can replace ict-patterns by wict-patterns or special patterns in order to compute dp-rank. Proposition 3.5. Let p(x) be a partial type and κ a cardinal number. The following are equivalent: 1. There is an ict-pattern for p of depth κ. 2. There is a wict-pattern for p of depth κ. 3. There is an special pattern for p of depth κ. Proof. 1⇒2 is obvious. 2⇒3. By a standard argument we may replace the array by a κ × ω-array with mutually indiscernible rows. 3⇒1. A compactness argument allows us to replace the array by a mutually indiscernible array of type κ×Z. Now we fix some witness b of the consistency of (3.1). Moreover we fix a row i ∈ κ. By deleting some positions we may assume the truth-value of ϕi (b, aij ) for j < 0 is constant. We may also assume the truth-value of ϕi (b, aij ) for j > 1 is constant. By replacing the formula ϕi (x, aij ) by ψi (x, aij aij+1 ) := ϕi (x, aij ) ↔ ¬ϕi (x, aij+1 ) one achieves a new pattern with negative values on the left (with a possible exception), negative values on the right (with a possible exception) and a positive value in the 0-th. column. More precisely the formula ψi (x, ui ) is ϕi (x, y i ) ↔ ¬ϕi (x, z i ) and the tuple is ci := ai2j ai2j+1 . By deleting at most two positions we obtain a row where ψ(b, cij ) is false for j 6= 0 and true for j = 0. As the new array is also mutually indiscernible, by point 3 of Remark 3.3 we have obtained an ict-pattern of type κ × Z for p. The following lemma allows us to avoid disjunctions in wict-patterns. It is well-known for ict patterns.
12
Proposition 3.6. Assume there is a wict-pattern( W for p with formulas S = ni ψik (x, y i ). Then for (ϕi (x, y i ) | i ∈ κ) and for each i ∈ κ ϕi (x, y i ) = k=0 κ some f ∈ ω with f (i) ≤ ni there is a wict-pattern for p(x) with formulas f (i) (ψi (x, y i ) | i ∈ κ). The same is true for ict-patterns. Proof. Let (aij )i∈κ,j∈ω be a mutually indiscernible κ × ω array showing there is a wict-pattern for p with formulas S. There is some b such that |= ϕi (b, ai0 ) and 6|= ϕi (b, aij ) for all i ∈ κ and j > 0. Hence, for some f ∈ ω κ with f (i) ≤ ni f (i) (b, ai0 ) for all i f (i) (ψi (x, y i ) | i ∈ κ) is
|= ψi
∈ κ. By point 2 in Remark 3.3, (aij )i∈κj∈ω together with
a κ × ω wict-pattern for p. The argument for ict-patterns is the same starting with an array of type κ × Z. Now, we see that for wict patterns, the same holds for conjunctions. Proposition 3.7. Assume there is a wict-pattern Vni for kp(x) with sequence of formulas S = (ϕi (x, y i ) | i ∈ κ) and ϕi (x, y i ) = k=0 ψi (x, y i ) for each i ∈ κ. Then for some f ∈ ω κ with f (i) ≤ ni there is a wict-pattern for p(x) with f (i) formulas (ψi (x, y i ) | i ∈ κ). Proof. If (aij )i∈κ,j∈ω is a mutually indiscernible κ×ω wict-pattern with sequence S there is some b such that |= ϕi (b, ai0 ) and 6|= ϕi (b, aij ) for all i ∈ κ and j 6= 0. Fix some row i. Since ϕi (b, aij ) is false for j > 0, there is some k ∈ {1, . . . ni } such that ψik (b, aij ) fails for infinitely many j > 0. By deleting elements in the row, we may assume that ψik (b, aij ) fails for all j > 0. Hence |= ψik (b, ai0 ) and 6|= ψik (b, aij ) for all and j 6= 0. Since this may be done for any i, we get f ∈ ω κ f (i)
such that |= ψi
f (i)
(b, ai0 ) and 6|= ψi
2 in Remark 3.3, (aij )i∈κ, wict-pattern for p.
(b, aij ) for all i ∈ κ and j 6= 0. By By point f (i)
j∈ω
together with (ψi
(x, y i ) | i ∈ κ) forms a κ × ω
Proposition 3.7 does not hold for ict-patterns as the following example shows. In the theory of dense linearly ordered sets without endpoints there is an ictpattern of depth 1 with formula y1 < x < y2 but there is no ict-pattern for none of the formulas y1 < x nor x < y2 . However we can reduce any conjunction to a conjunctions of at most two formulas in ict-patterns: Proposition 3.8. Assume there is an ict-pattern Vni for kp(x) with sequence of formulas S = (ϕi (x, y i ) | i ∈ κ) and ϕi (x, y i ) = k=0 ψi (x, y i ) for each i ∈ κ. Then for every i ∈ κ there is some Ii ⊆ {0, . . . , ni } withVat most two elements such that there is an ict-pattern for p(x) with formulas ( j∈Ii ψij (x, y i ) | i ∈ κ). Proof. If S together (aij )i∈κ,j∈Z is a mutually indiscernible κ × Z ict-pattern for p, there is some b such that |= ϕi (b, ai0 ) and 6|= ϕi (b, aij ) for all i ∈ κ and j 6= 0. Fix some row i. Since ϕi (x, aij ) is false for j > 0, there is some k ∈ {1, . . . ni } such that ψik (b, aij ) fails for infinitely many j > 0. By deleting elements in the 13
row, we may assume that ψik (b, aij ) fails for all j > 0. The same happens for for j < 0 with maybe another formula. These V two formulas provide the set Ii . V Hence, we get that |= j∈Ii ψij (b, ai0 ) and 6|= j∈Ii ψij (b, aij ) for all i ∈ κ and j 6= 0. By point 3 in Remark 3.3, (aij )i∈κ,j∈Z is the κ × Z array of an ict-pattern V with formulas ( j∈Ii ψij (x, y i ) | i ∈ κ). The following definition is standard, see [2] or [4] We include it here for a better readability of the paper. Definition 3.9. Let ϕ(x, y) be a partitioned formula. The dual-alternation number of ϕ(x, y) is the maximum number of changes of the truth-value of ϕ(b, ai ) (when i increases in O) for any tuple b and any indiscernible sequence (ai , i ∈ O). Remark 3.10. A (partitioned) formula ϕ(x, y) has dual-alternation number equal to zero iff {ϕ(M, a) | a ∈ M } is a finite set in any (some) model. We consider that a partitioned formula ϕ(x; y) with y the empty tuple has dual-alternation number zero. Proof. If the set {ϕ(M, a) | a ∈ M } is finite, in an indiscernible sequence (ai | i ∈ ω) some repetition must occur: ϕ(M, ai ) = ϕ(M, aj ) for some i 6= j hence ϕ(M, ai ) should be constant. This implies that the dual-alternation number of ϕ(x, y) is zero. Conversely, if the set {ϕ(M, a) | a ∈ M } is infinite, by a standard compactness plus Ramsey argument (see for instance Lemma 5.1.3 of [22]),one can construct an indiscernible sequence (ai | i ∈ ω) where ϕ(M, ai ) is not constant. Hence the dual-alternation number of ϕ(x, y) is at least one. Definition 3.11. We call a formula with dual-alternation number zero nonalternating. We will say that the formula is NA or a NA-formula. The following definition comes from Adler [3]. Definition 3.12. We call a set of partitioned formulas Ψ(x, y) an instantiable directed family (or a directed family for short) if for any two sets A, B defined by instances of formulas in Ψ either A ⊆ B, B ⊆ A, or A ∩ B = ∅. By an instance of ϕ(x, y) we mean the formula ϕ(x, b) for some b of the appropriate length. In the previous definition x is a finite tuple of variables, while we allow y to be an infinite tuple. Hence, although we use the same tuple y, not all formulas in Ψ must have the same finite tuple of parameter variables. Proposition 3.13. Let Ψ be a directed family of partitioned formulas. Then there is no wict-pattern with two different rows with formulas of Ψ or negation of formulas of Ψ (two formulas of the sequence with different index cannot both belong to Ψ ∪ ¬Ψ, where ¬Ψ denotes {¬ψ | ψ ∈ Ψ}). Obviously the same holds for ict or special patterns.
14
Proof. Assume there is a wict-pattern of depth 2 where the two formulas of the sequence or their negations belonging to Ψ. We may assume the pattern is of kind 2 × ω with mutually indiscernible rows. Let us denote by (Ai | i ∈ ω) the sets defined by the first row and (Bi | i ∈ ω) the sets defined by the second. We have to distinguish three cases depending on whether the formulas of the pattern or their negations belong to Ψ. Case 1. Both formulas belong to Ψ. By the consistency of the path (0, 0), A0 and B0 have nonempty intersection, thus without loss of generality A0 ⊆ B0 . By mutual indiscernibility A0 ⊆ B1 , which contradicts the consistency of the path (0, 0). Case 2. Only one formula belongs to Ψ. We may assume the Ai are defined by instances of formulas in Ψ while Bi = DiC is the complement of a set Di defined by an instance of a formula in Ψ. Again by the consistency of the path (0, 0), A0 ∩ B1C = A0 ∩ D1 is nonempty, hence either A0 ⊆ D1 or D1 ⊆ A0 . In the first case, by mutual indiscernibility, A0 ⊆ D0 and thus A0 ∩ B0 = ∅, which contradicts the consistency of the the path (0, 0). The second case implies B1C ⊆ A1 . This contradicts again the consistency of the path (0, 0) because C AC 1 ∩ B1 should be nonempty. Case 3. No formula belong to Ψ. In this case both Ai and Bi are the complements of Ci and Di respectively, sets defined by instances of formulas C in Ψ. Again the consistency of the path (0, 0) implies AC 1 ∩ B1 = C1 ∩ D1 is nonempty. Hence we may assume C1 ⊆ D1 and thus B1 ⊆ A1 . By mutual indiscernibility, B0 ⊆ A1 which contradicts again the consistency of the path (0, 0). Definition 3.14. Let λ be a cardinal (finite or infinite). We say that a complete theory T is λ-VCA if there is collection of λ instantiable directed families hΨi (x, y) | i ∈ λi such that each definable 1-set Sin the single variable x is a boolean combination of instances of formulas in i∈λ Ψi (x, y) and instances of NA-formulas of kind ψ(x, y). We call a complete theory T VCA-minimal if it is 1-VCA. Obviously any VC-minimal theory is VCA-minimal. However the converse fails as Proposition 4.4 shows. Fact 3.15. Let Σ(x, y) be a set of formulas without parameters, where the tuple y can be infinite. Assume each definable set with parameters with free variables among x is definable by an instance of some formula in Σ. Then, for each formula ϕ(x, y) without parameters there is a finite subset Θ of Σ such that each instance of ϕ(x, y) is an instance of some formula in Θ. Proof. T ∪ {¬∃z∀x(ϕ(x, y) ↔ ψ(x, z)), ψ(x, y) ∈ Σ} is inconsistent. By compactness T ∪ {¬∃z∀x(ϕ(x, y) ↔ ψ(x, z)), ψ(x, y) ∈ Θ} is inconsistent for some finite Θ ⊆ Σ. Corollary 3.16. Any VCA-minimal theory is dp-minimal. 15
Proof. Let Ψ(x, y) be a directed family witnessing the VCA-minimality of the theory. We may assume that each formula in any wict-pattern for x = x is a boolean combination of formulas of Ψ and NA-formulas: in each row of the pattern some boolean combination provided by Fact 3.15 will occur infinitely many times. By Propositions 3.6 and 3.7 we can assume each row of a wictpattern is constituted by a formula of Ψ, a negation of a formula of Ψ or a non-alternating formula (the negation of an NA-formula is NA). By Proposition 3.13 Ψ can contribute with at most one row, and no non-alternating formula can occur in a wict-pattern. We will see in Proposition 4.4 that any dp-minimal ordered Abelian group is VCA-minimal. In fact, this is more general: Proposition 3.17. If T is λ-VCA then the dp-rk(T ) ≤ λ. Proof. As in Corollary 3.16 using propositions 3.7, 3.6, 3.13 and the fact that NA-formulas cannot occur in a wict-pattern.
16
4
dp-minimal ordered groups
Here we work again with the monster model of some complete theory. The following is not the standard definition of the dual VC-density, but equivalent to it. See [4]. Definition 4.1. The dual VC-density of a partitioned formula ϕ(x, y), denoted by vc∗ (ϕ(x, y)), is the infimum of all real numbers r > 0 such that for any r finite set A of |y|-tuples |S ϕ (A)| = O(|A| ). Here S ϕ (A) denotes the set of all maximally consistent sets of formulas of kind ϕ(x, a) or ¬ϕ(x, a) with a ∈ A. For more details on the dual VC-density, see [4]. In the proof of next proposition we use the following facts: Facts 4.2. 1. |S ϕ (A)| coincides with the number of atoms of the Boolean algebra of sets generated by the definable sets ϕ(C, a) with a ∈ A. Here C denotes the monster model (or any model) of the theory and ϕ(C, a) denotes the set defined by ϕ(x, a) in C. 2. Let B be a Boolean algebra and B1 , B2 be given finite subalgebras (closed under ∧ and ¬) of B. Let us denote by at(Bi ) the number of atoms of Bi . Then the number of atoms of the subalgebra generated by B1 ∪ B2 is at most at(B1 )at(B2 ). 3. Let A1 , . . . , An be a directed family of sets, i.e. if Ai ∩ Aj 6= ∅ then either Ai ⊆ Aj or Aj ⊆ Ai . Then the Boolean algebra of sets generated by A1 , . . . , An has at most n + 1 atoms. Proposition 4.3. In a VCA-minimal theory any formula ϕ(x, y) has dual VCdensity at most 1. Proof. By fact 3.15, for each formula ϕ(x, z) there is a finite set Θ of formulas of kind µ(x, u) which are Boolean combinations of formulas in Ψ and NA-formulas, such that each instance of ϕ(x, z) is equivalent to an instance of some formula in Θ. Let Ψ0 denote respectively the set of formulas from Ψ occurring in the boolean combinations of formulas in Θ and let Υ be the set of NA-formulas occurring in the boolean combinations of formulas in Θ. Let N be a common upper bound of the number of different sets each formula in Υ can define. We may assume all formulas in Ψ, Θ and Υ have the same parameter variables, say u. Given a set of z-parameters A, we can choose a set of u-parameters B of size at most |A| such that each instance of ϕ(x, z) with parameters from A is an instance of some formula in Θ with parameters from B. Hence, any definable set ϕ(C, a) with a ∈ A is a boolean combination of sets of kind ψ(C, b) where ψ(x, u) ∈ Ψ ∪ Υ and b ∈ B. |Υ| Now it is not difficult to see that |Sϕ (A)| ≤ (|Ψ0 | |A| + 1)2N . This holds because the boolean algebra generated by the sets defined by Ψ0 -formulas with parameters from B has at most |Ψ0 | |A| + 1 atoms. And the Boolean algebra 17
generated by Υ-formulas with parameters in B has at most 2N there are at most N |Υ| such nonequivalent formulas.
|Υ|
atoms becase
Here is the complete characterization of all dp-minimal ordered groups. We say that an ordered Abelian group has finite dimension iff dimp (G) is finite for all prime p. Proposition 4.4. Let G be an ordered group. The following are equivalent: 1. G is VCA-minimal. 2. Any formula ϕ(x, y) has dual VC-density at most 1 (in the theory of G). 3. G is dp-minimal. 4. G is Abelian and has finite dimension. Proof. 1 implies 2 By proposition 4.3. 2 implies 3. By [7] Proposition 3.2. 3 implies 4. G is abelian by Proposition 3.3 of [21]. Assume now G is Abelian but dimp (G) is infinite for some prime p. As [G : pG] is infinite, let (bj )j∈ω be an infinite set of positive elements non congruent modulo pG. Let a be an element greater than any bj (in the monster model!). Then the following is a wict-pattern of depth 2: (x > ipa | i ∈ ω), (x ≡ bi mod p | i ∈ ω). 4 implies 1. Now assume that for each prime number p, dimp (G) is finite. Then, by Remark 5 in 2.2, each p-regular rank should be finite. By Theorem 2.4, any formula can be written as a boolean combination of formulas of the following kind: nx
≤ t(y)
nx
≡
nx
≡
t(y) t(y)
(4.1) mod ∆
(4.2) m
mod ∆ + p G
(4.3)
where n ∈ Z, ∆ ∈ RJp (G), p ∈ P, m ≥ 1 and t(y) is a term. Since the formulas of kind (4.1) and (4.2) define convex subsets, they are boolean combination of definable initial segments. So we may express any formula as a boolean combination of definable initial segments and formulas of kind (4.3). As the formulas defining initials segments constitute a directed family it only remains to prove that formulas of kind (4.3) are NA. Claim 4.5. Every nonempty instance of the formula nx ≡ t(y) mod ∆ + pm G is a coset of ∆ + pr G, where n = n0 ps , gcd(n0 , p) = 1 and r = min {m − s, 0}. Proof. easy. Finally
18
Claim 4.6. If dimp (G) is finite, the formula nx ≡ t(y)
mod ∆ + pm G
is NA. r Proof. By claim . 4.5, it suffices to show [G : ∆ + p G] is finite. Since G/∆ p(G/∆) ' G/(∆ + pG) is a free Fp -module of rank dimp (G/∆) . then G/∆ pr (G/∆) ' G/(∆+pr G) is a free Z pr Z-module of rank dimp (G/∆)
(see [15]). This implies [G : ∆ + pr G] = pr dimp (G/∆) ≤ pr dimp (G) , which is finite.
Observe that Corollary 3.16 gives another proof that 1 implies 3. We cannot replace VCA-minimal by VC-minimal. In [8] it is shown that any convexly orderable ordered Abelian group is divisible. Moreover any VCminimal theory is convexly orderable (see [9] for a proof of this). This shows that there are many VCA-minimal ordered Abelian groups non convexly orderable.
19
5
Chain conditions
We start with Lemma 5.1. It is a result about pure groups and it is a generalization of the Chinese Remainder Theorem to the non-Abelian case. We assume there is some group G, a, b are elements of G and H is a subgroup of G. We use the congruence notation to work with left cosets: a ≡ b mod H means aH = bH. If H1 and H2 are subgroups of G, we let H1 H2 denotes the following set: {h1 h2 | h1 ∈ H1 , h2 ∈ H2 }. Obviously H1 H2 is a subgroup iff H1 H2 = H2 H1 . Observe that the condition (5.1) depends on the order of the sequence of groups, while the conclusion not. Lemma 5.1. Let G be a group, H1 , . . . , Hn a sequence of subgroups of G satisfying the following: ! \ \ (Hi Hr ) = Hi Hr for r = 2 . . . n (5.1) i
